Periodicity of Hyperbolic Cosine

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Theorem

Let $k \in \Z$.

Then:

$\cosh \left({x + 2 k \pi i}\right) = \cosh x$


Proof

\(\displaystyle \cosh \left({x + 2 k \pi i}\right)\) \(=\) \(\displaystyle \frac {e^{x + 2 k \pi i} + e^{- \left({x + 2 k \pi i}\right)} } {2 i}\) Definition of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{i \left({-i x + 2 k \pi}\right)} + e^{i \left({i x + 2 \left({-k}\right) \pi}\right)} } {2 i}\) $i^2 = -1$ and simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{i \left({-i x}\right)} + e^{i \left({i x}\right)} } {2 i}\) Periodicity of Complex Exponential Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^x + e^{- x} } {2 i}\) $i^2 = -1$
\(\displaystyle \) \(=\) \(\displaystyle \cosh x\) Definition of Hyperbolic Cosine

$\blacksquare$


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